13:33:44.140449 arp who-has xxx.xxx.xxx.xxx tell xxx.xxx.xxx.xxx
13:33:44.141428 arp reply xxx.xxx.xxx.xxx is-at 00:d0:bb:db:06:01
but i am not in the central point i am on an computer in the network that i like to span my network for supecious extivity in a different ip address range.

Can you explain a bit more with an simple example for example looking at host xxx.xxx.xxx.123 to understand better.

I like to monitor from my computer other computers connected to
our network but on an differnet ip address range and i am
sitting on an linux computer.

I have access to the units on an different ip address range and i
hope that i do not have to be the place where all internet
traffic is passing on the way out to the internet where the
cisco router is located in order to intercept traffic and try to
figure out if an computer connected to our network is
compromised by worms or virus that could cause unwanted
bandwidth usage making problems for other clients worse ...

So i am sitting on an linux computer that i like to
start an script that runns the check on the network ...

Sounds like nmap is not the right tool for your requirement. IPS is what you are looking for. Snort comes into my mind if you want to use open sourced solution.
If port scanning is your way of identifying illegal network activity, you must be able to identify which ports (out of 65K range) are legitimately opened. I think it's quite a challenge to do this if you have many servers and running many applications.
On the second thought, if your servers have been hardened and only a minimum number of ports are opened, then maybe port scanning will do the job. For example, if you have a server which is running a web server, you only expect port 80 (and some other common network services) to be opened. If port 21 suddenly shows up in the result of nmap scan (but it is not supposed to be running FTP service), then that should trigger an alert.

IMHO try to see (and to use) NTop, a very good free and open source
tool for network traffic assessment.
It can show network traffic sorted according to various criteria and
can do network assets discovery and categorization according to their
OS and users. These are only two example but NTop can do much more
things...
I've used it (with success) in the past!

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